# Regular category

In category theory, a regular category is a category with finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain exactness conditions. In that way, regular categories recapture many properties of abelian categories, like the existence of images, without requiring additivity. At the same time, regular categories provide a foundation for the study of a fragment of first-order logic, known as regular logic.

A category C is called regular if it satisfies the following three properties:[1]

In a regular category, the regular-epimorphisms and the monomorphisms form a factorization system. Every morphism f:X→Y can be factorized into a regular epimorphism e:X→E followed by a monomorphism m:E→Y, so that f=me. The factorization is unique in the sense that if e':X→E' is another regular epimorphism and m':E'→Y is another monomorphism such that f=m'e', then there exists an isomorphism h:E→E' such that he=e' and m'h=m. The monomorphism m is called the image of f.

A functor between regular categories is called regular, if it preserves finite limits and coequalizers of kernel pairs. A functor is regular if and only if it preserves finite limits and exact sequences. For this reason, regular functors are sometimes called exact functors. Functors that preserve finite limits are often said to be left exact.

Regular logic is the fragment of first-order logic that can express statements of the form

which is natural in C. Here, R(T) is called the classifying category of the regular theory T. Up to equivalence any small regular category arises in this way as the classifying category of some regular theory.[2]

A regular category is said to be exact, or exact in the sense of Barr, or effective regular, if every equivalence relation is effective.[4] (Note that the term "exact category" is also used differently, for the exact categories in the sense of Quillen.)